In fact, this construction works for an arbitrary non-linear connected Stein group \(G\) . Recall that the linearizer of a complex Lie group is the intersection of the kernels of all holomorphic representations. So it is contained in the nilpotent radical. By [1]}, the linearizer contains a copy of \(\mathbb {C}^\times \)  as a direct factor. Consider the case when the linearizer is isomorphic to \(\mathbb {C}^\times \) . For example, this holds for the Heisenberg group. (It is worth mentioning that it can be easy to deduce from [2]} that any distorted copy of \(\mathbb {C}^\times \)  is contained in the linearizer.) Then there is a length function exponentially distorted on it. It holds by the same argument as above with the only difference that we need to show that \(G/\mathbb {T}\)  is \(\mathbb {R}\) -linear. This can be done as in [1]} using the fact that \(G/\mathbb {T}\)  is an extension of an \(\mathbb {R}\) -linear group by \(\mathbb {R}\) . The general case can be reduced to this by factorization.

